This research is on the design of experiment and statistical inference for quantal responses. Its primary application is in bioassay where the probability of response depends on the amount of stimulus or drug given to a living organism. The formulation and solution of the bioassay problems will be by Bayesian methods. Specific objectives will include a study of prior distributions applicable to bioassay, formulation of a design criterion and guidelines for selecting experimental designs in terms of an experimenter's subjective opinion, and numerical methods for examining and summarizing posterior distributions of various characteristics of quantal response curves. Particular emphasis will be placed on estimating the extreme percentage points of quantal response curves. Concrete examples will be used to motivate and demonstrate the applicablity of prior distributions as a formal means of expressing experimental uncertainties.